# Seminar 3 Monopoly. Simona Montagnana. Week 25 March 20, 2017

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1 Seminar 3 Monopoly Simona Montagnana Week 25 March 20, 2017

2 2/41 Question 2 2. Explain carefully why a natural monopoly might occur. a Discuss the problem of using marginal cost pricing to regulate a natural monopoly and suggest some possible solutions.

3 3/41 Concepts Natural Monopoly is a monopoly created and sustained by economies of scale over the relevant range of output for the industry. When does Natural Monopoly occur? For instance when the industry presents very high fixed costs, very low marginal cost, due to the nature of the production process (when there is an expensive technology or a distribution network such as electricity, railways, water, gas, etc.). Natural monopolies often occur because of economies of scale, when the production of a good or service is more efficient if it is carried out by a single company rather than divided among several firms with the same technology of the monopolist (figure 1): because one firm can meet the market demand at a lower cost than two or more firms can.

4 4/41 Figure: 1 - Long Run Average Curve Put simply, with one firm producing Q 3 the average cost is 1 per unit, with three firms, each one produce less output, and the average cost per unit is higher.

5 5/41 In terms of demand curve and cost curves, a natural monopoly occurs when the quantity at which the marginal cost curve intersects the demand curve is below the minimum efficient scale - MES: the level of output at which the average cost curve has a minimum (figure 2) Figure: 2 - Minimum Efficient Scale

6 6/41 In general the monopolist has the following profit-maximization problem: max Π = max [R(Q) C(Q)] Q Q The first order condition for this problem requires is: Π Q = 0 R (Q) C (Q) = 0 R (Q) = C (Q). Which can be written as: MR(Q) = MC(Q)

7 7/41 Alternatively, the maximisation problem can be expressed as: max Q Π = max [P(Q)Q C(Q)] Q In this case the first order condition for this problem is: MR = P(Q) + Q P (Q) = C (Q) = MC

8 Remember that for a downward sloping demand, price decreases with quantity, P (Q) < 0, and since MR = P(Q) + Q P (Q), the MR curve lies beneath the inverse demand curve P(Q). Figure: 3 - Natural Monopoly and welfare loss 8/41

9 9/41 From the figure 3 we can see that there are two solutions for the FOC. To know which one is the maximum we have to look at the second order condition: 2 Π Q 2 0 R (Q) C (Q) 0 R (Q) C (Q) MR (Q) MC (Q). The maximum point where this condition is verified is Q, the one where MC and MR cross and the slope of the latter is lower (or the marginal revenue curve crosses the marginal cost curve from above).

10 10/41 The second order condition in the alternative approach is as follows: 2P (Q) + P (Q)Q C (Q) 0 Note that: at level Q, we have a welfare loss with respect to the efficient quantity Q E, that we obtain where the MC curve crosses the inverse demand P(Q). The welfare loss is measured as the difference between the value lost for unfulfilled demand between Q E and Q (area below the inverse demand P(Q) between Q E and Q ), and the cost saved by the monopolist for not producing up to Q E (area below the MC curve between Q E and Q ).

11 11/41 In order to prevent the welfare loss, the government might decide to regulate the industry. There are two possibilities: the average cost pricing, when we fix P = AC(Q); the marginal cost pricing, when we fix P = MC(Q).

12 12/41 2.b. Discuss the problem of using marginal cost pricing to regulate a natural monopoly and suggest some possible solutions. The marginal cost pricing means to set a price equal to the marginal cost P = MC. In terms of quantity, this occurs at the efficient level Q E where the inverse demand crosses the MC curve. The problem is that at price P(Q E ) = MC(Q E ) the monopolist incurs in a loss, measured as the difference between the price MC(Q E ) and the costs AC(Q E ), multiplied by the quantity Q E (area in red in figure below).

13 13/41 Figure: 4 - Marginal cost pricing and monopolist loss

14 14/41 Solutions: For the monopolist the problem is that the marginal cost pricing MC(Q E ) will not produce enough revenues to cover the total cost (P(Q E ) < AC). Put differently, the efficient level of output Q E is not affordable for the firm, and the monopolist would prefer to go out of business. The marginal cost price regulation is the best choice for the consumers (the social welfare is maximized when P = MC), since we get the lowest price and the highest quantity, but not for the firm. That is why usually natural monopolies are regulated or operated by governments: to keep the firm operating the government pays a subsidy (called lump - sum subsidy) to the firm in order to cover the economic loss.

15 15/41 Question 3 3. Using a model of a price discriminating monopoly, answer the following: a. What is perfect price discrimination? Would perfect price discrimination be socially preferable to uniform pricing? Justify. b. What is second degree price discrimination? Give an example. c. What is third degree price discrimination and what condition is required for the monopolist to be able to perform it? If the monopolist faces two different markets, show that it will tend to charge a lower price in the market with the more elastic demand.

16 16/41 (a) First degree price discrimination: selling each unit of a good at a different price. The perfect price discrimination or first-degree price discrimination occurs when the monopolist sells different units of output for different prices and these prices might differ for each person (often infeasible). This is possible because: The monopolist has perfect information about the consumers/markets. The prices are different to each consumer and each unit. A monopoly with first-degree or perfect price discrimination is an efficient market.

17 17/41 a. Would perfect price discrimination be socially preferable to uniform pricing? Justify. The perfect price discrimination can be socially preferable to uniform pricing, depending on the social criteria adopted by the society. Social criteria are a normative concept. If a social planner only cares about the total level of welfare and does not consider distributive issues (in this case concerning monopolist and consumers), then the perfect price discrimination is socially preferable: the monopolist firm - having perfect information about the consumer s marginal willingness to pay (or reserve price)- can capture all consumer surplus by pricing each unit exactly at the consumer s marginal willingness to pay. In this particular case the monopolist obtains the entire social surplus, and so profit maximization is synonymous with maximizing social welfare.

18 18/41 For example: a doctor in a small village, who charges different fees to patients - the same service is sold to different consumers at different prices (in function of the willingness to pay or reserve price of each consumer). With FDPD the monopolist makes much higher profits, because takes the entire consumer surplus.

19 19/41 Price B D A C (Pm) E Consumer Surplus (ABC) F Monopolist Surplus (BCED) Welfare loss CFE) MC Qm Output Figure: 5 - UniformPrice

20 20/41 Price A I (p 1 ) G p 1 is the willingness to pay for the first consumer Consumer Surplus (GHC) B (p i ) H C Monopolist Surplus (BCED+IGBH) Welfare loss D(p n ) E F MC Q 1 Q i Output 1 Figure: 6 - Imperfect discrimination

21 21/41 Price A I(p 1 ) (p 2 ) (p 3 ) (p 4 ) B(p i ) D(p n ) G Different prices p for each consumer depending by their willingness to pay C E Monopolist Surplus F MC Q i Q n Output 1 Figure: 7 - Perfect discrimination

22 22/41 Exercise A monopolist has a cost curve C(Q) = cq and an inverse demand function p(q) = a Q, where a > 0 and c > 0. a Calculate the MC(Q) and MR(Q). b Which is the price p that maximizes his profit? Calculate the profit in this case. Suppose the monopolist applies a first degree price discrimination. c Which is now the level of output? Calculate the monopolist profit in this case.

23 23/41 Uniform price monopoly: the monopolist sells Q the quantity of a good and P single price. max Q Π = max Q [P(Q)Q C(Q)] - Profit maximization problem MR = MC - Necessary condition of this problem MR = P(Q) + QP (Q) α Q + Q( 1) α 2Q - marginal revenue function So we get: α 2Q = c, then Q = α c 2 and P = α+c 2

24 24/41 With numbers: Suppose, we have α = 500 and c = 100 P(Q) = 500 Q C(Q) = 100Q MC = 100 MR = 500 2Q Inverse demand function Cost function Marginal cost Marginal revenue With uniform pricing, the monopolist maximises the profit when MC(Q) = MR(Q) MC(Q) = MR(Q) 100 = 500 2Q Q = 200 and p = 300 The monopolist profit is: Π = 40, 000

25 25/41 Now, suppose the monopolist is able to practice a first degree price discrimination, that means he knows the consumer s marginal willingness to pay, and he sets different prices for each consumer, and the price (or willingness to pay) for the last consumer will be exactly equal to: p = MC, so p = c Then the level of output will be: Q = α c Exactly where p = MC, this result is not surprising because the monopolist with first degree price discrimination obtains all the consumers surplus, and he will produce the efficient level for the economy.

26 26/41 With first degree price discrimination, the monopolist maximises the profit: Π = (α c)q 2 With numbers: The efficient level of output is: Q = α c = = 400 The monopolist profit is Π = ( )400 2 = 80, 000

27 27/41 (b) Second degree price discrimination: setting a different price for each buyer of the good. The second degree price discrimination (non linear pricing) occurs when the monopolist sells different units of output for different prices, but every individual who buys the same amount of the good pays the same price. The prices differ across the units of good, but not across people. Some useful examples are: Bundle of products purchased: e.g. different menu packages, cable TV (Sky), train or airlines tickets. Amount purchased (nonlinear pricing): e.g. sizes of grocery products 2X 1 (buy 2 and pay 1): P(1 + 2) < P 1 + P 2.

28 28/41 The monopolist knows that it faces different individuals with different demand functions but it cannot tell who is who. The consumers adopt a self-selection approach. In this case the monopolist has incomplete information about the consumers and is not able to extract the consumer surplus. He must use self-selection devices to set the right price-quantity or price-quality packages. Example: An Airline (or train) pricing The monopolist firm wants to price the tickets according to type (business or tourist seats) but does not know the willingness to pay of each consumer. The monopolist sets prices for business and tourist seats so that consumers self-select.

29 29/41 Suppose we have: 2 consumers: one is business (B) and one is tourist (T) with different willingness to pay and different utility functions u B and u T. The prices tickets are p F and p C respectively for First Class (F) and Coach seats (C). Participation constraints are expressed in terms of willingness to pay for each consumer: u B (F ) p F 0 u T (C) p C 0 Self-selection constraints for each consumer are: u B (F ) p F u B (C) p C u T (C) p C u T (F ) p F

30 30/41 We can assign numbers to the different willingness-to-pay values: First Class Coach u B (F ) = 1000 u B (C) = 400 u T (F ) = 500 u T (C) = 300 The self-selection and participation constraints read: 1000 p F 400 p C Consumer B buys First Class ticket (self-selection constraint) 300 p C 500 p F Consumer T buys Coach ticket (self-selection constraint) 1000 p F 0 Consumer B decides to travel (participation constraint) 300 p C 0 Consumer T decides to travel (participation constraint)

31 31/41 The solutions for the monopolist are obtained as follows: From the last inequality the monopolist charges the maximum price to the Tourist type consumer, pc = 300, so as to maximise profits but still making sure he travels. By substituting pc=300 in the first two inequalities, we get: p F 500 to prevent the Tourist type consumer from choosing the First Class, p F 900 to prevent the Business consumer to travel choosing the Coach. Actually, if the monopolist could know the types, he would charge the Business consumer type up to 1000, from his participation constraint, in order to maximise profits. But this is not publicly available information, and from the self-selection constraint we see this consumer can be charged at most 900.

32 32/41 The reason why we start from a binding participation constraint of the low-demand consumer to find the solution relies on the assumption of single-crossing utility curves for the two consumers types (they intersect only in one point). From this it follows that for the low demand consumer the participation constraint is binding (the one from which we start to find the solution), while for the high-demand consumer the self-selection constraint is binding. The proof of this result is quite involving.

33 33/41 (c)third degree price discrimination: setting a different price for each class of buyer. The third degree price discrimination occurs when the monopolist sells units of output to different groups of people at different prices, but every unit of output sold to a given group is sold at the same price. The monopolist, using some signals (age, location, profession, etc?), can separate the markets. An example might be student discounts and senior citizen discounts tickets at museums. Consider a model where a monopolist is able to sell a good in two markets (alternatively say two consumers groups) 1 and 2. In particular the monopolist can identify two different groups of people with different inverse demand curve and can sell a good or a service at different prices.

34 34/41 Say P 1 (Q 1 ) and P 2 (Q 2 ) are inverse demand functions respectively for group 1 and 2. The profits maximization is: MaxΠ(Q 1 Q 2 ) = MaxP 1 (Q 1 ) + P 2 (Q 2 ) c(q }{{} 1 + Q 2 ) }{{} revenues costs The necessary conditions for this problem are: { MR1 (Q 1 ) = MC 1 (Q 1 + Q 2 ) MR 2 (Q 2 ) = MC 2 (Q 1 + Q 2 )

35 35/41 The elasticity of demand is ε = Since R = PQ, we can write: Q Q P P = Q P P Q MR = ϑr ϑq = ϑp ϑq Q + P = ϑp ϑq Q P P + P = [ MR = P 1 + ϑp ] [ Q = P ] [ = P 1 1 ] ϑq P ε ε from which we see how the Marginal revenue depends on elasticity. The FOC for this problem can written as: { P1 (Q 1 ) + P (Q 1 )Q 1 = C (Q 1 + Q 2 ) P 2 (Q 2 ) + P (Q 2 )Q 2 = C (Q 1 + Q 2 )

36 36/41 If the market 2 has a more elastic demand then market 1, we can write: 1 1 ε 2 (Q 2 ) > ε 1 (Q 1 ) < ε 2 (Q 2 ) ε 1 (Q 1 ) 1 1 < 1 1 ε 2 (Q 2 ) ε 1 (Q 1 ) We can say that the monopolist will charge a lower price in market 2 and P 1 > P 2 (where the consumers are more sensitive to the pricing) [ ] P 2 (Q 2 ) }{{} smaller 1 1 ε 2 (Q 2 ) }{{} larger = c In general remember also that C (Q) = ϑc(q) ϑq = c

37 37/41 We get: p C (Q) p = 1 ε = 1 ε Figure: 8 - Elasticity of demand

38 38/41 Exercise A monopolist knows the inverse demand curve of two different groups of consumers: p 1 (Q 1 ) = 200 4Q 1 p 2 (Q 2 ) = 140 Q 2 Suppose the monopolist applies a third degree price discrimination, and his marginal cost is constant c = 40. Find the prices for two different groups of consumers ( p 1 and p 2 ) and show that the monopolist will charge a lower price in the market with the more elastic demand.

39 39/41 Total revenue from the 1 group: TR 1 = p(q 1 )Q 1 = (200 4Q 1 )Q 1 = 200Q 1 4Q 2 1 Marginal revenue from the group 1: MR 1 = ϑtr 1 ϑq 1 = 200 8Q 1 Because MR 1 = MC 1 and MC = 40 MR 1 = ϑtr 1 ϑq 1 = 200 8Q 1 = 40 = MC We get Q 1 = 20 and P 1 = 120

40 40/41 Similarly for group 2 TR 2 = p(q 2 )Q 2 = (1400 Q 2 )Q 2 = 140Q 2 Q 2 2 MR 2 = ϑtr 2 ϑq 2 = 140 2Q 2 Because MR 1 = MC 1 and MC = 40 MR 2 = ϑtr 2 ϑq 2 = 140 2Q 2 = 40 = MC We get Q2 = 50 and P 2 = 90 The monopolist applies a lower price to the consumer group with higher elasticity of demand. P 2 = 90 < P 1 = 120 p MC p = 1 ε ε 2 = 9 5 > 3 2 = ε 1

41 41/41 Readings Varian (2014), Intermediate Microeconomics: A Modern Approach (Ninth Edition), Ch. 25 and 26.